WASHINGTON/MIAMI BEACH, Fla. Republican Donald Trump called U.S. President Barack Obama and Democrat Hillary Clinton the "co-founders" of Islamic State, ratcheting up his assertion that they are responsible for the rise of the militant group and sparking renewed criticism of his leadership ability.

Clinton's White House campaign on Thursday called the remarks a "false claim," in her latest response to a series of attacks by Trump in which he has sought to portray America as less safe, blame Democrats and depict himself as the only one who can restore security.

Democrats, in turn, have used Trump's often hyperbolic statements ahead of the Nov. 8 election to argue he is unfit to be president and lacks the temperament to be trusted with matters of national security.

"This is another example of Donald Trump trash-talking the United States," senior policy adviser Jake Sullivan said in a statement.

"What's remarkable about Trump's comments is that once again, he's echoing the talking points of (Russian President Vladimir) Putin and our adversaries to attack American leaders and American interests, while failing to offer any serious plans to confront terrorism or make this country more secure," Sullivan said.

For Republicans uncertain about whether Trump has the discipline to stick to an attack against Clinton, the latest comments were concerning. Many see the New York real estate mogul as spending too much time fighting within his own party and have called on him to refocus his campaign message on Clinton.

"ISIS is a solid GOP message to show contrast with Hillary Clinton and the failures of the Obama-Clinton administration," said Alice Stewart, a Republican strategist who remains undecided about the nominee, using acronyms for Islamic State and the Republican Party.

But, she added, "Trump should have simply said that the Obama administration's decision to pull all troops out of Iraq, with no stay-behind agreement, created a vacuum and allowed ISIS to metastasize. It's absurd for him to say that Obama and Clinton are founders of ISIS - and he can't blame the media for this."

A group of about 70 Republicans, including five former members of Congress, called on the Republican National Committee to stop helping Trump in the wake of his recent remarks and instead focus on getting members of Congress re-elected.

"Trump’s divisive and dangerous actions are not only a threat to our other candidates, but to our party and the nation," the letter stated.

Some Republicans see a small silver lining in Trump talking more about Clinton.

"It is helpful – at least to the rest of the ticket – that he is focussing a little more on Clinton than on other Republicans, whether defeated primary opponents or other elected officials who are on the ballot, for a change," said former New Hampshire Republican Chairman Fergus Cullen, who is not supporting Trump.

"But tomorrow, or later today, he could blame (Republican Senator) Jeff Flake for A-Rod’s retirement," Cullen said, referring to Yankees player Alexander Rodriguez's decision to leave professional baseball. "I have zero confidence in Trump’s ability to stay on one message or to drive one message for any length of time longer than about 10 seconds."

CRITICISM OF IRAQ WAR

Trump has previously criticized Clinton for supporting the Iraq War in 2003 while she was a U.S. senator. Trump frequently says, in contrasting himself with Clinton, that he opposed the war - but in interviews before the invasion he did voice support.

Now, Trump is arguing that in trying to end the war and withdrawing U.S. troops in 2011, Clinton, who was secretary of state at the time, and Obama created Islamic State.

Republicans frequently trace the birth of Islamic State to the Obama administration’s decision to withdraw the last U.S. forces from Iraq by the end of 2011.

But many analysts argue its roots lie in the decision of George W. Bush’s Republican administration to invade Iraq in 2003 without a plan to fill the vacuum created by Saddam Hussein’s ouster. It was Bush’s administration, not Obama’s, that negotiated the 2009 agreement that called for the withdrawal of all U.S. forces from Iraq by Dec. 31, 2011.

Clinton posted on social media website Twitter that Trump's comments are disqualifying.

"Anyone willing to sink so low, so often should never be allowed to serve as our commander-in-chief," she wrote.

The White House declined to comment on Trump's claim.

Appearing in Miami Beach, Florida, on Thursday morning, Trump repeated his attack for the third time, saying the U.S. government "has unleashed ISIS."

"In fact, I think we’ll give Hillary Clinton ... most valuable player," Trump said. "ISIS will hand her the most valuable player award. Her only competition is President Barack Obama."

Trump first made the assertion in a speech on Wednesday night in Florida, saying, "I call them co-founders" of Islamic State.

In an interview on Thursday morning, Trump defended the remarks.

"Is there something wrong with saying that?" Trump told CNBC. "Why - are people complaining that I said he was the founder of ISIS? All I do is tell the truth, I'm a truth teller."

Trump was also asked by radio host Hugh Hewitt if he "meant that (Obama) created the vacuum, he lost the peace."

"No," Trump responded. "I meant he's the founder of ISIS. I do."

The Democratic National Committee lambasted Trump's remarks. "Donald Trump should apologise for his outrageous, unhinged and patently false suggestions on the founding of ISIS," the DNC said in a statement. "This is yet another out of control statement by a candidate who is unravelling before our very eyes."

Former New York City Mayor Rudy Giuliani defended Trump on CNN, saying his remarks were “legitimate political commentary.”